Self-Organized Fission-Fusion Control Algorithm for Flocking …Complexity T ˘ˇ : Parametersinthesimulation. Parameters ˆ˝ $ tar& - C " esc tar F esc Value . . . . . m m . −10

Research ArticleSelf-Organized Fission-Fusion Control Algorithm forFlocking Systems Based on Intermittent Selective Interaction

Panpan Yang Maode Yan Jiacheng Song and Ye Tang

School of Electronic and Control Engineering Changrsquoan University Xirsquoan 710064 China

Correspondence should be addressed to Panpan Yang panpanyangchdeducn

Received 29 June 2018 Revised 19 December 2018 Accepted 22 January 2019 Published 11 February 2019

Academic Editor Alex Alexandridis

Copyright copy 2019 Panpan Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In nature gregarious animals insects or bacteria usually exhibit paradoxical behaviors in the form of group fission and fusionwhich exerts an important influence on grouprsquos pattern formation information transfer and epidemiology However the fission-fusion dynamics have received little attention compared to other flocking behavior In this paper an intermittent selectiveinteraction based control algorithm for the self-organized fission-fusion behavior of flocking system is proposed which bridges thegapbetween the two conflicting behaviors in a unified fashion Specifically a hybrid velocity coordination strategy that includes boththe egalitarian and selective interactions is proposed where the egalitarian interaction is to maintain the flockrsquos order and achievethe fusion behavior while the selective interaction strategy is for the response to external stimulus information and generates thefission behavior Numerical simulations demonstrate that the proposed control algorithm can realize the self-organized fission-fusion behavior of flocking system under a unified frameworkThe influences of the main control parameters on the performanceof the fission-fusion behavior are also discussed In particular the trade-off parameter 120572 balances the exploration (fission) andexploitation (fusion) behaviors of flocking system and significantly enhances its movement flexibility and environmental adaptivity

1 Introduction

In nature gregarious animals insects or bacteria often aggre-gate into a cohesive group to gain some survival advan-tages such as reducing predation risk improving foragingefficiency and saving individual energy [1ndash4] Within thesegrouping species group formation is usually a highlydynamic process group size and composition may changefrequently during the life time of members by group splittingormerging which is usually referred to as the ldquofission-fusionrdquobehavior [5 6] (see Figure 1)

The term ldquofission-fusionrdquo was firstly introduced by HansKummer [7] to describe the social system of a few taxa ofnonhuman primates such as chimpanzees geladas andhamadryas baboons that change the size of their groupsby means of the fission and fusion of subunits (calledparties or subgroups) [8] As a matter of fact fission-fusionbehavior is a commonly seen phenomenon in nature Forinstance Bechsteinrsquos bats usually aggregate into a large groupduring pregnancy and lactation for thermoregulation andsplit into smaller subgroups in the post-lactation period

[9] European starlings often gather into a giant swarm toenhance the possibility of finding food and rapidly segregateinto separated subunits in the presence of predatorrsquos attack[3] On the other hand the fission-fusion behavior is alsoof practical significance for some artificial flocking systemsFor example a group of unmanned aerial vehicles (UAVs)can gather into a cohesive flock to carpet bomb enemyrsquostargets and split into smaller clusters to avoid the attack ofanti-aircraft fireThe autonomous underwater vehicle (AUV)swarm that is executing the seabed exploration task mayaggregate into a small group to go across the narrow tunneland expand to its original state to monitor the missionarea Therefore investigation on the fission-fusion behaviorof flocking system is of both theoretical significant andapplication value

Over the last decades scientists have been working vig-orously on the underlying mechanisms of flocking behaviorIn [10] a famous flocking model (namely ldquoBoidsrdquo) withthree heuristic rules (cohesion separation and alignment)was proposed to animate the bird flocking behavior Latera simplified discrete-time flocking model for self-propelled

HindawiComplexityVolume 2019 Article ID 2187812 12 pageshttpsdoiorg10115520192187812

2 Complexity

(a) (b)

(c) (d)

Figure 1 Typical fission-fusion behaviors of flocking system in nature (a) European starlings Sturnus vulgaris over Rome [3] (b) shoal ofsardines in Kwazulu-natal South Africa (c) Merinos drsquoArles sheep Ovis aries [6] (d) segregation in mixed cocultures of primary goldfishkeratocytes (PFK red) and EPC fish keratocytes (EPC green) [18]

particles considering the alignment rule only was introducedin [11] In addition a zonal model was proposed in [12]which is able to produce some typical collective behaviors(such as swarm torus dynamic parallel and highly dynamicparallel group) by adjusting certain parameters In generalthese worksmainly deploy the egalitarian interaction methodand focus on the group fusion aspect little attention has beenpaid to the fission behaviorHow a randomly distributed flockforms a coherent group and splits into multiple subgroups ina unified manner remains largely unknown

Recently owing to the advancements in the high-resolution spatiotemporal flocking data acquisition and anal-ysis techniques more and more lines of evidence havedemonstrated that the selective interaction rather than theegalitarian interaction is more effective in the collectiveresponse of flocking system [13 14] A selective interactionbased hierarchical structure was found in pigeon flockrsquosdirectional choice which was proved to bemore flexible thanthe egalitarian strategy [15] In [16] a hybrid control architec-ture combining both compromise (egalitarian interaction)and leadership (selective interaction) was proposed to drivethe grouprsquos movement decisions A route-dependent switchmechanism between hierarchical and egalitarian strategies inpigeon flocks was found in [17] which revealed that pigeonstend to follow the average of neighbors while moving along asmooth trajectory and select a certain leader to follow whensudden turns or zigzags occurs Selective interaction whichimplements the information transfer by following a specificleader in an implicit manner shows significant potential inthe fission behavior of flocking system

Inspired by the above empirical evidences an intermit-tent selective interaction based control algorithm is proposedfor the self-organized fission-fusion behavior of flockingsystem The main contributions of this paper are as follows

(i) A specifically designed intermittent selective inter-action behavior is integrated into the conventionalaverage velocity consensus scheme and endows theflock the capability of spontaneous splitting in thepresence of external stimulus information

(ii) Aweight adjustment strategy is designed to automati-cally balance the egalitarian interaction and intermit-tent selective interaction in different environmentswhich guarantees the stability and environmentaladaptability of flocking system

(iii) A dynamic threshold value for the fission behavior isdesigned based on the order parameter of individualswhich make the flock more flexible in the groupfission-fusion behavior

The rest of this paper is organized as follows In Section 2the coordinated control problem for the self-organizedfission-fusion behavior of flocking system is proposed and thepitfall of the conventional velocity consensus based egalitar-ian interaction in flocking control is analyzed In Section 3a unified fission-fusion control framework is established byintroducing the intermittent selective interaction into theegalitarian interaction scheme and a dynamic weight balancestrategy is designed for the group fusion and fission behaviorNumerical simulations are provided in Section 4 to verify

Complexity 3

the effectiveness of the proposed control algorithm and someconcluding remarks and future work are drawn in Section 5

2 Problem Formulation

Consider a flocking system consisting of 119873 identical mem-bers each individual is governed by the following doubleintegrator dynamics

119909119894 = V119894

V119894 = 119906119894 (1)

where 119909119894 isin 119877119899 is the position vector of individual 119894 V119894 isin119877119899 denotes its velocity vector and 119906119894 isin 119877119899 represents theacceleration vector (control input) acting on it

Constrained by the limited sensing ability each membercan only communicate with its nearby neighbors within aspecific range [20] Hence the neighboring set of individual 119894can be denoted by

119873119894 (119905) = 119895 119889119894119895 le 119877 119895 = 1 2 sdot sdot sdot 119873 119895 = 119894 (2)

where 119889119894119895 = 119909119894 minus 119909119895 denotes the Euclidean distancebetween individual 119894 and 119895 and119877 is the sensing radius of eachindividual

Generally speaking the collective behavior of flockingsystem is generated by both the local interactions with itsnearby neighbors and the external environment [1] Conse-quently the control framework for the self-organized fission-fusion behavior can be roughly formulated as

119906119894 = 119906in119894 ( sum119895isin119873119894(119905)

119906 (119909119894 119909119895 V119894 V119895)) + 119892119894119906out119894 (3)

where 119906in119894 denotes the internal force exerting on individual119894 from its nearby neighbors 119906(119909119894 119909119895 V119894 V119895) is a specificallydesigned function for the interaction force between indi-vidual 119894 and 119895 and 119906out119894 is the external force from theenvironment In addition 119892119894 is deployed to show whetherindividual 119894 is influenced by external information If 119892119894 = 1we say that the motion of individual 119894 is governed by both itsnearby neighbors and surrounding environment otherwise119892119894 = 0 and it is only influenced by its neighbors

Remark 1 Owing to the limited sensing ability only a smallportion of individuals (eg that lie on the edge of the flock)can directly sense the external stimuli and are influencedby the external force 119906out119894 while the motion of others ismerely governed by the internal force 119906in119894 from their nearbyneighbors [13] Therefore the internal force (also called thelocal interaction rule) between individuals plays a key role inthe self-organized fission-fusiondynamics of flocking system

According to the existing literature most of the internalinteractions between individuals follow the cohesion align-ment and separation rules [19] which can roughly be imple-mented via the following position and velocity coordinationterm

119906in119894 = 119906pos119894 + 119906vel119894 (4)

Here the position coordination term 119906pos119894 usually deploysan artificial potential function with long distance attractionand short distance repulsion properties ie

119906pos119894 = sum119895isin119873119894(119905)

2( 11988610038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817 minus11988710038171003817100381710038171003817119909119894 minus 119909119895100381710038171003817100381710038173) (5)

where 119886 119887 gt 0 are parameters that determine the strength ofthe attraction and repulsion force respectively

In addition 119906vel119894 is the velocity alignment term thatguarantees the order of flocking system

119906vel119894 = minus sum119895isin119873119894(119905)

(V119894 minus V119895) (6)

Equation (6) is the famous ldquovelocity consensusrdquo algorithmwhich iswidely used in the control protocol design of flockingbehavior for its mathematical elegance and simplicity [21]However such information consensus property (also calledthe egalitarian strategy) although very effective in the fusionbehavior gives a flock the tendency of group cohesion andhampers the process of splitting [22] which may degrade itsmovement flexibility and reaction rapidity especially in thepresence of external stimulus such as multiple food source orpredatorrsquos threat [16]

Motivated by this fact we are trying to develop a novelcontrol framework for the self-organized fission-fusionbehavior of flocking system which aims to realize the spon-taneous fusion behavior in free space and the reactive fissionbehavior under external stimuli in a unified fashion Inparticular this framework is the supplement of the traditionalegalitarian strategy which can deal with the external stimulusin a more flexible and efficient way

3 Self-Organized Fission-FusionControl Algorithm Based onIntermittent Selective Interaction

In this section a unified control algorithm for the self-organized fission-fusion behavior of flocking system is pro-posed by introducing an intermittent selective interactioninto the traditional egalitarian interaction framework whichpromotes the stimulus information transfer within the flockand makes it more sensitive to environmental variation

31 Intermittent Selective Interaction Based Control Frame-work for the Self-Organized Fission-Fusion Behavior Theself-organized fission-fusion behavior of flocking system isvirtually composed of two competing parts where the fusionbehavior requires members to form a highly coherent andordered group while the fission behavior needs to break upthe coherence and split intomultiple smaller subgroups [6 8]

In order to represent the two seemingly contradictorybehaviors in a unified manner an intermittent selective inter-action is integrated into the velocity coordination termTogether with the traditional egalitarian interaction basedvelocity coordination approach the control framework for

4 Complexity

the self-organized fission-fusion behavior of flocking systemcan be generalized as

119906119894 = 119906pos119894 + 120572119906ega119894 + (1 minus 120572) 119906sel119894⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119906vel119894

+ 119892119894119906out119894 (7)

It is obvious that the control input 119906119894 is composed of threeparts

(1) The position coordination term 119906pos119894 with the char-acteristics of long distance attraction and short distancerepulsion follows (5) to coordinate the spatial distribution ofindividuals

(2) The velocity coordination term 119906vel119894 is to regulate thealignment of individuals Here a hybrid velocity alignmentstrategy which combines both the egalitarian and selectiveinteraction based velocity coordination is proposed via aweighted parameter 120572

119906vel119894 = minus120572 sum119895isin119873119894(119905)

(V119894 minus V119895)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119906ega119894

minus (1 minus 120572) (V119894 minus V119897119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119906sel119894

(8)

where V119897119894 is the velocity of the temporary leader that individ-ual 119894 selects and 120572 isin [0 1] is the parameter adjusting theweight between 119906ega119894 and 119906sel119894

(3)The external stimuli 119906out119894 have a variety of forms suchas the attraction to a known food source or repulsion to anobstacle or predatorrsquos threat [23] (the detailed form of 119906out119894will be given in Section 4)

Remark 2 In (8) the egalitarian interaction based velocitycoordination term 119906ega119894 drives individuals towards their aver-age velocity to form a highly ordered and cohesive flock Onthe contrary the selective interaction based velocity coordi-nation term 119906sel119894 makes it follow the velocity of the temporaryleader which thus endows individual 119894 the potential of split-ting from the flock Specifically by adjusting the weightingparameter 120572 the flock can exhibit the self-organized fusion-fission behavior according to the environmental variationadaptively

Remark 3 It is also worth noting that in (8) the temporaryleader of individual 119894 is time varying and evolves accordingto some leader selection rule which is fundamentally differentfrom the traditional leader-follower approach as our methodis to promote the external stimulus information transferwithin the flock in an implicit manner

In the following we will investigate the temporary leaderselection rule from the perspective of promoting externalstimulus information transfer within the flock

32 Selective Interaction Based Temporary Leader-FollowerRelationship As has been discussed in Section 2 the self-organized fission-fusion behavior of flocking system is thedirect (by directly sensing the external stimulus information)or indirect (through collective propagation of changes inbehaviors by group members) response to external stimuliin which the information flow between individuals plays asignificant role [24]

In the free motion of flocking system the informa-tion flow is mainly conducted by averaging all the nearbyneighborsrsquo information [25] However it is suggested to bemore efficient to switch to the single neighbor interactionmode when responding to some external stimuli with abruptacceleration or sudden turning [17] Inspired by this fact aselective interaction rule is proposed from the perspectiveof maximizing the external information transfer within theflock

119897119894 = max119862119894119895 119862119894119895 gt 119862lowast 119895 isin 119873119894 (119905) (9)

where119862119894119895 denotes the influence of neighbor 119895 on individual 119894and 119862lowast is the threshold value for the selective interaction

Remark 4 It can be observed form (9) that individual 119894 willselect themost influential neighbor 119897119894 as the temporary leaderwhich is a widely seen phenomenon in nature as individualstend to follow the aged experienced or strong neighbors toenhance the opportunity of survival [5 8]

In the fission behavior of flocking system individuals arerequired to select the proper temporary leader to promotethe stimulus information transfer more efficiently From thebiological visual perception mechanism [24] it is known thatindividuals are usually very sensitive to the rapid variationof nearby neighbors On the other hand members that aremoving with fast maneuvering often carry more externalstimulus information [14] Hence the influence of neighbor 119895on individual 119894 can be described as

119862119894119895 = 120577119894119895 110038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198601

∙ 120581119894119895 (V119894 ∙ V119895)1003817100381710038171003817V1198941003817100381710038171003817 10038171003817100381710038171003817V11989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198602

(10)

where1198601 and1198602 are the influence of neighbor 119895 on individual119894 with respect to position and velocity and 120577119894119895 and 120581119894119895 arethe coefficients of the position and velocity related influencerespectively

Additionally a threshold value of 119862lowast is designed toprevent the unexpected fission behavior such as stochasticdisordered splitting

119862lowast = 119890minus120573120593119894 (11)

where120573 gt 0 is the threshold value adjustment parameter and120593119894 is the order parameter of individual 119894120593119894 = 1119873119894 + 1

10038171003817100381710038171003817100381710038171003817100381710038171003817119873119894sum119895=1

V11989510038171003817100381710038171003817V1198951003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (12)

with119873119894 being the number of nearby neighbors

Remark 5 It can be seen in (11) that the specifically designedthreshold value 119862lowast is a monotone decreasing function withrespect to the order parameter 120593119894 When 120593119894 997888rarr 0 the flockmoves in a disordered state the threshold value119862lowast is high andthe selective interaction behavior is not likely to happen andthe flock tends to perform the fusion behavior when 120593119894 997888rarr

Complexity 5

1 the flock is in an ordered state and the threshold value119862lowast is small and the selective interaction behavior is easy tooccur which will facilitate the external stimulus informationtransfer and induce the fission behavior

4 Simulation Studies

In this section various numerical demonstrations are pro-vided to verify the feasibility and effectiveness of the proposedfission-fusion control algorithm

41 Performance Metrics In order to evaluate the per-formance of the proposed control algorithm in the self-organized fission-fusion behavior of flocking system a seriesof metrics are firstly defined as follows

(1) Polarization Index (120595)Thepolarization index120595 representsthe velocity alignment degree of all the individuals in the flock

120595 = 11198731003817100381710038171003817100381710038171003817100381710038171003817119873sum119894=1

V1198941003817100381710038171003817V11989410038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (13)

where119873 is the number of all the individuals in the flock andV119894 denotes the velocity of individual 119894

It can be seen from (13) that the polarization indexvaries between [0 1] If the velocities of all the individualsin the flock are aligned 120595 asymp 1 otherwise if the velocity ofindividuals are completely randomly distributed 120595 asymp 0Remark 6 It should be noted that the polarization index 120595seems a little like the order parameter 120593119894 defined in (12) Infact120595 can be seen as the global order parameter of the wholeflocking system while 120593119894 only demonstrates the local orderparameter of individual 119894Remark 7 The polarization index can well reflect the velocityvariation of individuals in the group fusion process In thegroup fission process the velocity of individuals tends todiverge and the polarization index will decline However thedeclining of polarization index is not always correspondingto the fission behavior Therefore based on the concept ofbimodality coefficient in statistical analysis theory [26] anew performance metric namely differentiation index isintroduced

(2) Differentiation Index (120582) The differentiation index 120582demonstrates the velocity divergence degree of individuals inthe flock

120582 = S2 + 1K

(14)

where

S = (1119873)sum119873119894=1 (V119894 minus V)3radic((1119873)sum119873119894=1 (V119894 minus V)2)3 (15)

K = (1119873)sum119873119894=1 (V119894 minus V)4((1119873)sum119873119894=1 (V119894 minus V)2)2 minus 3 (16)

HereS andK denote the skewness and kurtosis of individ-ualsrsquo velocity distribution respectively

Remark 8 According to the bimodality coefficient in statisti-cal analysis theory we can conclude from (14) to (16) that thedifferentiation index which reflects the velocity distributionof individuals also varies between [0 1] To be specificwhen 120582 lt 59 the velocity of individuals is in unimodaldistribution Specifically when 120582 = 59 the velocity ofindividuals is in uniform distribution When 120582 gt 59the velocity of individuals is in bimodal distribution whichdemonstrates that the velocity of individuals tends to divergeIn particular when 120582 = 1 the velocity of individuals is inBernoulli distribution and the velocity is completely divergedTherefore the differentiation index 120582 is an effective metric toevaluate the fission performance of flocking system

42 Group Fusion and Fission Experiment Suppose 50 indi-viduals lie randomly within a 15 times 15m2 region and theirinitial velocities are all zero The sensory range of eachindividual is 119877 = 5m A group formation (fusion) andmultitarget tracking (fission) scenario is carried out to showthe entire process of group fusion and fission

Initially members in the flock aggregate from theirinitial distributions and move in formation with a consistentvelocity [5 5]ms under the control protocol (7) Two targetslie on the moving path of the flock symmetrically and theirinitial positions are [30 35]Tm and [35 30]Tm respectively

Two members (labelled as individuals 1 and 2) thatfirstly sense the targets will initiate the fission behavior bytracking the motion of targets towards opposite directionsThe tracking force follows

119906out119894 = minus120574 [(119909119894 minus 119909tar119894) + (V119894 minus Vtar119894)] (17)

where 119909tar119894 and Vtar119894 are respectively the position and velocityof target 119894 and 120574 gt 0 is the gain of the tracking force

Two targets which are also governed by (1) remainstationary until the flock gets into their sensing radius If theflock approaches the targets the escaping behavior will beactivated which follows

119906esctar = 119888esctar

119878sum119895=1

exp(minus10038171003817100381710038171003817119909tar119896 minus 11990911989510038171003817100381710038171003817119897esc ) 119896 = 1 2 (18)

where 119878 = 119895 119909tar119896 minus 119909119895 le 119877tar 119895 = 1 2 sdot sdot sdot 119873 is the setof flock member within the sensing radius of target 119896 119897esc gt 0is the correlation length and 119888esctar is the gain of the escapingbehavior

The other simulation parameters are listed in Table 1 andthe simulation results are shown in Figures 2 and 3

Figure 2 is the trajectory of the flocking system in thefusion-fission process To be specific Figure 2(a) shows theinitial distribution of the flock and targets where mem-bers in the flock lie randomly in the predefined regionand the two targets lie on the moving path of the flocksymmetrically Figure 2(b) gives the flockrsquos fusion processwhere individuals with random distribution aggregate intoa coherent group and organize into a highly ordered

6 Complexity

Table 1 Parameters in the simulation

Parameters 119886 119887 120577119894119895 120581119894119895 120573 120572 120574 119897esc 119877tar 119888esctar

Value 20 3 05 2 25 05 123 15m 3m 55

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(a)

minus10

0

10

20

30

40

50

60

70

y (m

)0 10 20 30 40 50 60 70minus10

x (m)

(b)

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(c)

0 10 20 30 40 50 60 70minus10x (m)

minus10

0

10

20

30

40

50

60

70

y (m

)

(d)

Figure 2 Trajectory of individuals in the fusion-fission processThe blue hollow circles denote the initial positions of individuals black solidcircles demonstrate their final positions green lines represent the trajectories of individuals red solid circles are the position of two targetsand gray solid circles are the individuals that firstly sense the targets

formation to move towards the targets Figure 2(c) shows thebeginning of the fission behavior when the flock encounterstwo targets the targets execute the escaping behavior oncethey detect the flock and the two members that firstly sensethe targets try to split form the flock to track the targetsseparately and then the flock tends to split into two subgroupsunder the fission-fusion control algorithm (7) Figure 2(d)illustrates the final position of the flock and the targets wherethe coherent flock ultimately splits into two subgroups totrack their corresponding targets independently

Figure 3 demonstrates the velocity variations of individ-uals during the group fusion-fission process in both 119883 and

119884 axis It can be seen that during 119905 = 0 sim 4s the flock is inthe fusion phase and the velocities of all individuals tend toconverge to the same value After that the fission behavioris activated and the velocity of members tends to divergewhich will ultimately converge to that of the moving targets(the velocities of the two targets are marked in red and blackrespectively)

Figure 4 shows the polarization index and differentiationindex variation in the group fusion and fission process fromwhich one can clearly see that in the group fusion processthe polarization index 120595 increases from a very small value to1 in finite time which suggests that the flock aggregates from

Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y

Figure 3 Velocities of individuals in the fusion-fission process

20 6 84 10t (s)

0

05

1

Figure 4 The polarization index and differentiation index in thefusion-fission process

random distribution to an ordered state and the velocitiesof individuals are well aligned The differentiation index 120582on the other hand fluctuates around 05 and the velocityof individuals is in unimodal or uniform distribution After119905 = 4s the flock fission process begins and the polarizationindex 120595 tends to decrease due to the velocity divergence ofindividualsThedifferentiation index120582 begins to increase andultimately converges to 1 which illustrates that the velocity ofindividuals is in bimodal distribution and the fission behaviorcompletes

From the above simulation results one can clearly see thatthe proposed fission-fusion control algorithm (7) can makethe flock aggregate into a coherent group in free space andsegregate into multiple subgroups when a small portion ofmembers split from the flockTherefore the proposed controlalgorithmguarantees both the order and flexibility of flockingsystem which is much more universal than the traditionalflocking control method and also very crucial for the survivaland evolution of flocking system

43 Comparison with Conventional Egalitarian Strategy Inorder to demonstrate the superiority of the proposed con-trol algorithm (7) a comparative simulation is carried outwith a typical consensus based control method proposedin [19] where the velocity of members in the flock is

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

Figure 5 Trajectory of individuals under the control algorithm of[19]

coordinated according to (6)The other settings are the sameas Section 42

Figure 5 demonstrates that the consensus based controlmethod can also achieve the fusion behavior where scat-tered distributed individuals aggregate into a highly orderedformation However this approach is unable to realize thefission behavior when two targets move towards oppositedirections Only the two individuals that sense the targets cansplit from the flock and track the moving targets the othermembers still move along their original path This is largelydue to the inherent coherence property of the consensusbased velocity coordination term which hampers the flockrsquossplitting process and degrades its efficiency in response toexternal stimuli

Figure 6 shows the velocity of individuals in both119883 and119884axis where green lines are the velocity curves of individualsthat are not directly influenced by external stimuli and redand black lines are the velocity curves of individual 1 andindividual 2 respectively It can be clearly seen that membersin the flock firstly aggregate into a coherent group underthe consensus based control scheme and their velocitieseventually converge to the same value However whenexternal stimuli act on partial members of the flock onlytwo individuals split from the flock and the rest of theflock remain unchanged and keep their original movementdirection

Figure 7 shows the polarization index and differentiationindex variation under the control algorithm of [19] In thegroup fusion process both the polarization index and differ-entiation index variation are the same as those of Figure 4which illustrates that the control algorithm of [19] can wellaccomplish the fusion behavior In the group fission processthe polarization index 120595 remains about 1 (the whole flock isgenerally in an ordered state) However the differentiationindex is not significantly increased (fluctuates around 055)which implies that the velocity of individuals is still inunimodal distribution and thus the fission behavior does not

8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)

Figure 6 Velocity of individuals under the control algorithmof [19]

0

05

1

2 4 6 8 100t (s)

Figure 7Thepolarization index anddifferentiation index under thecontrol algorithm of [19]

occur Therefore the control algorithm of [19] cannot realizethe self-organized fission behavior in the presence of externalstimuli

Compared with the fusion-fission control algorithm pro-posed in this paper the traditional velocity consensus basedcontrol method hampers the external information transferwithin the flock and is unable to realize the self-organizedfission behaviorwhenflock encounters some external stimuliTherefore our approach is more efficient in promoting localstimulus information transfer and makes the flock moreadaptable to the dramatically changing environment

44 The ReunionRejion Capability of Flocking System Thesimulation in Section 42 demonstrates the self-organizedfusion and fission capabilities of flocking system Howeverwhether the separated subgroups can reunite into a singleflock still remains unknown Here an additional simulationis carried out to verify the reunion capability of the flock

The two targets are supposed to disappear at 119905 = 8sAfter that an additional navigational term is incorporatedinto algorithm (7) with the following form


+ 119906nav119894 + 119892119894119906out119894 (19)

where 119906nav119894 = minus984858(V119894 minus Vnav119894) 119894 = 1 2 with 984858 being thenavigational gain and Vnav119894 being the navigational velocity ofsubgroup 119894

0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)

Figure 8 The reunion capability of the flocking system aftersplitting

Here we let 984858 = 10 Vnav1 = [minus5 5]ms andVnav2 = [5 minus5]ms The navigational term 119906nav119894 drives thetwo subgroups to move close to each other Once they arewithin the sensing range of any individuals in the flock thenavigation termdisappears and the flock begins to implementthe reunion behavior based on the local interactions only

Figure 8 shows the reunion behavior of the flocking sys-tem after splitting which suggests that the proposed fission-fusion coordination algorithm (7) can make the separatedsubgroups reunite into a coherent single flockThis is becausewhen two targets disappear the external stimuli acting on theflock no longer exist and the navigational term drives thetwo subgroups to move close to each other Once they arewithin the sensing range of any individuals in the flock thenavigation term disappears and individuals determine theirmotion based on the local interaction with nearby neighborsonly Meanwhile there is no abrupt turning or accelerationof any single individual and thus the selective interactionbehavior will not occur whichmakes the selective interactionbased velocity coordination term lose efficacy in the fission-fusion control algorithm (7) Therefore they will execute thegroup fusion law and regather into a single flock as long asthey are in the sensing range of each other

45 Discussion The Influence of Parameters on the Fission-Fusion Behavior As seen in Table 1 there are many parame-ters in the fission-fusion control algorithmof flocking systemwhich together determine the performance of fission-fusionbehavior In order to better illustrate the influence of themainparameters on the fission-fusion behavior of flocking systema detailed discussion is given as follows

451 The Influence of Parameters 119886 and 119887 on Flockrsquos DensityThe parameters 119886 and 119887 in the artificial potential function (5)is to determine the relative distance between individuals inthe steady state From (5) we know that 1198890 = radic119887119886 is theequilibrium distance between the attraction and repulsion

Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)

Figure 9 The polarization index of flocking system with different120572

forces If 119889119894119895 gt radic119887119886 the attraction force will be dominantwhich drives individuals to move towards each other and theflock size will shrink On the contrary if 119889119894119895 lt radic119887119886 therepulsion forcewill be dominate and individuals tend tomoveaway from each other meanwhile the group size will expandThereforeradic119887119886 is relevant to the density of flocking system

On the other hand if the equilibrium distance is largerthan the sensing range 119877 of each individual ie radic119887119886 gt 119877members in the flock are unable to get the information ofneighbors and the fusion behavior cannot be guaranteedTherefore the feasible range of parameters 119886 and 119887 shouldsatisfy radic119887119886 isin (0 119877]452The Influence of Parameter120572 on the Polarization Index120595It is known that the parameter 120572 in (8) plays a significant rolein the fusion-fission control of flocking system as it balancestheweights of egalitarian interaction and selective interactionin the velocity coordination of individuals Here an extensiveinvestigation is carried out to demonstrate the influence ofparameter 120572 on the polarization index 120595 of fission-fusionbehavior

Figure 9 shows the polarization index variation withdifferent 120572 in the flock fission-fusion process where the flockis expected to execute the fusion behavior from 119905 = 0 sim 4s andperform the fission behavior afterwards We can see that if 120572is high (eg 120572 = 1) the polarization index increases rapidlyfrom 01 sim 09 which shows that the flock aggregates fromrandom distribution to an ordered state and the formationprocess is accomplished However the polarization indexstill maintains a high value (about 09) when the fissionbehavior begins suggesting that the flock still moves in acoherent formation and the group fails to achieve the fissionbehavior With the decline of 120572 the polarization index in thefission process decreases synchronously which shows that the

=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)

Figure 10Thedifferentiation indexof flocking systemwith different120577119894119895

fission behavior tends to happen If 120572 further decreases thepolarization index will keep a low value in the fusion processand group fusion behavior is not likely to happenTherefore amoderate120572 (about 05) canmake the flock perform the fusionand fission behavior simultaneously and endows the flock theability of implementing the fusion behavior in free space andspontaneous fission behavior under external stimuli

From the above results it can be concluded that theparameter 120572 realizes the trade-off of exploration (externalstimulus information) and exploitation (the existing infor-mation in flocks) of flocking system in the group fission andfusion process As 120572 increases the ldquoexploitationrdquo behavioris highlighted and the flocking system tends to use itsinternal information to aggregate into a coherent groupOn the contrary with the decrease of 120572 the ldquoexplorationrdquobehavior is strengthened and the flock tends to split via theexternal information In nature exploration and exploitationare a very commonly seen phenomenon in flocking systemwhere exploration enhances the possibility of finding food ordecreases the rate of being prey while exploitation maintainsthe original motion of flocking system and keeps its stabilityTherefore the method proposed in this paper is consistentwith the characteristic of natural flocking system which ismore adaptable than the traditional flocking control method

453 The influence of Parameters 120577119894119895 and 120581119894119895 on the Differenti-ation Index 120582 The parameters 120577119894119895 and 120581119894119895 are the coefficientsof the position and velocity related influence in (10) whichdetermine the temporary leader to follow and influence thefission behavior Here a comprehensive study is performedto show the influence of parameters 120577119894119895 and 120581119894119895 on thedifferentiation index 120582

From Figure 10 we can see that the differentiation indexis in bimodal distribution (the fission behavior emerges)if the coefficient of the position related influence term

10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)

Figure 11The differentiation index of flocking systemwith different120581119894119895

120577119894119895 isin [035 1] Specifically the fission behavior is wellaccomplished if 120577119894119895 = 05 (the differentiation index 120582 asymp 1and the velocity of individuals is in Bernoulli distribution)If 120577119894119895 lt 05 (eg 120577119894119895 = 035) the fission behavior can also beachieved although the differentiation index 120582 is less than 1However if 120577119894119895 continues to decline the differentiation index120582 tends to be less than 59 the velocity of individuals isin unimodal distribution and the fission behavior does notoccur On the other hand if 120577119894119895 gt 05 we can see that 120582 gt 59Although the velocities of individuals are diverged it may notguarantee a successful fission behavior because 120582 is much lessthan 1

Based on the above discussion we can conclude thatrelatively moderate position correlation is beneficial forthe fission behavior of flocking system Too large positioncorrelation will make individuals follow the motion of itsnearest neighbor which may neglect the stimulus informa-tion hidden in the neighborhood and cause the failure offission behavior However if the position correlation is toosmall the fission behavior may also fail because the flockmay be in a disordered state without considering the positiondistribution of nearby neighbors

In Figure 11 it is observed that the fission behavior isvery sensitive to the variation of the coefficient of the velocityrelated influence term 120581119894119895 If 120581119894119895 is small (eg 120581119894119895 le 2)the fission behavior can be accomplished However withthe increase of 120581119894119895 the differentiation index 120582 declinesdramatically which cannot achieve the fission behavior offlocking system The main reason lies in that too strongvelocity correlation maymake individuals too sensitive to thevelocity variation of nearby neighbors and hence the orderand stability of the flocking system cannot be guaranteed

Therefore a relatively small 120581119894119895 can make individualsobtain appropriate velocity correlation with nearby neigh-bors meanwhile maintaining the order of the flock

=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10


454The Influence of Parameter 120573 on the Polarization Index120595andDifferentiation Index 120582 As is shown in (11) the thresholdvalue 119862lowast for the fission behavior is time varying along withthe variation of the order parameter 120593119894 where the adjustmentparameter 120573 is to determine the steepness of the thresholdvalue curve In the following numerical simulations areperformed to show the influence of parameter 120573 on thepolarization index 120595 and differentiation index 120582

Figures 12 and 13 show the polarization index and differ-entiation index variation with different 120573 It can be clearlyobserved that when 120573 is small (eg 120573 lt 1) the polarizationindex120595 increases from 0 to 1 rapidly whichwell accomplishes

Complexity 11

the fusion behavior Meanwhile the differentiation index 120582fluctuates around 05 suggesting that the velocity of individ-uals is in unimodal distribution However the polarizationindex and differentiation index still keep unaltered in thegroup fission process which illustrates the failure of fissionbehaviorTherefore too small 120573 is unable to realize the fissionbehavior The main reason lies in that when 120573 is smallthreshold value curve is complanate and 119862lowast maintains a highvalue which prevents the occurrence of the fission behavior

From Figures 12 and 13 we can also see that with theincrease of 120573 (1 lt 120573 lt 2) the group fusion behavior can beachieved (ie 120595 asymp 1 120582 asymp 05) In the group fission process(119905 gt 4s) the polarization index120595 tends to declinemoderatelywhile the differentiation index 120582 begins to increase (120582 gt 59)which implies that the velocities of individuals are in bimodaldistribution and hence the fission behavior occurs

Moreover if 120573 continues to increase the threshold valuecurve becomes steep and119862lowast tends to declineThus the fissionbehavior becomes easy to occur and the flock tends to splitin a stochastic fashion Therefore the order of the flock isnot well maintained and both the fusion and fission behaviorcannot be accomplished in an ordered way

5 Conclusions and Future Work

This paper investigates the self-organized fission-fusion con-trol problem of flocking system A hybrid velocity regulationmechanism which combines the selective interaction basedvelocity coordination into the traditional average consen-sus rule is proposed to endow a flock the capability ofaggregating in free space and spontaneous splitting in thepresence of external stimulus Various numerical simulationsdemonstrate the feasibility and effectiveness of the proposedmethod in flocking fusion-fission control This paper bridgesthe gap between flock fission and fusion behavior which isexpected to provide new insight into the coordinated controlof flocking system

In this paper there are many parameters in the self-organized fission-fusion control algorithm of flocking systemand we conduct the parameter selection procedure in a trial-error method Although the influences of the main parame-ters on the performance of the fission-fusion behavior are dis-cussed in detail how to select the appropriate parameter set inan automatic and efficient way remains largely unknown Inthe future we will investigate the control parameter selectionmethod via someoptimization techniques such as the geneticalgorithm

Data Availability

The data used to support the findings of this study have notbeen made available because another paper is in preparationbased on these data

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (61803040) China PostdoctoralScience Foundation (2018M643556) and the Fundamen-tal Research Funds for the Central Universities of China(300102328403)

References

[1] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[2] D Grunbaum ldquoAlign in the sandrdquo Science vol 312 no 5778 pp1320ndash1322 2006

[3] I L Bajec andFHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[4] A Marrocco H Henry I B Holland M Plapp S J Seror andB Perthame ldquoModels of self-organizing bacterial communitiesand comparisons with experimental observationsrdquo Mathemat-ical Modelling of Natural Phenomena vol 5 no 1 pp 148ndash1622010

[5] I D Couzin ldquoBehavioral ecology Social organization infission-fusion societiesrdquoCurrent Biology vol 16 no 5 pp R169ndashR171 2006

[6] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[7] H Kummer ldquoSocial behavior and habitatrdquo Science vol 174 p179 1971

[8] F Aureli C M Schaffner C Boesch et al ldquoFission-fusiondynamics new research frameworksrdquo Current Anthropologyvol 49 no 4 pp 627ndash654 2008

[9] I Pretzlaff G Kerth and K H Dausmann ldquoCommunallybreeding bats use physiological and behavioural adjustmentsto optimise daily energy expenditurerdquoNaturwissenschaften vol97 no 4 pp 353ndash363 2010

[10] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquoComputer Graphics vol 21 no 4 pp 25ndash341987

[11] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[12] I D Couzin J Krause R James GD Ruxton andN R FranksldquoCollective memory and spatial sorting in animal groupsrdquoJournal of Theoretical Biology vol 218 no 1 pp 1ndash11 2002

[13] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Acadamy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[14] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquoEURASIP Journal onAdvances in Signal Processing vol2012 no 1 pp 1ndash17 2012

[15] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[16] X-K Xu G D Kattas and M Small ldquoReciprocal relationshipsin collective flights of homing pigeonsrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 85 no 2Article ID 026120 2012

12 Complexity

[17] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 pp 5805-5805 2014

[18] EMehes andTVicsek ldquoSegregationmechanisms of tissue cellsfrom experimental data to modelsrdquo Complex Adaptive SystemsModeling vol 1 no 1 pp 1ndash13 2013

[19] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[20] N W F Bode D W Franks and A J Wood ldquoLimited interac-tions in flocks Relating model simulations to empirical datardquoJournal of the Royal Society Interface vol 8 no 55 pp 301ndash3042011

[21] W Ren ldquoOn consensus algorithms for double-integratordynamicsrdquo IEEE Transactions on Automatic Control vol 53 no6 pp 1503ndash1509 2008

[22] M Liu P Yang X Lei and Y Li ldquoSelf-organized fission controlfor flocking systemrdquo Journal of Robotics vol 2015 Article ID321781 10 pages 2015

[23] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[24] A Strandburg-Peshkin C R Twomey N W F Bode et alldquoVisual sensory networks and effective information transfer inanimal groupsrdquo Current Biology vol 23 no 17 pp R709ndashR7112013

[25] D Sumpter J Buhl D Biro and I Couzin ldquoInformation trans-fer in moving animal groupsrdquoTheory in Biosciences vol 127 no2 pp 177ndash186 2008

[26] D W Muller and G Sawitzki ldquoExcess mass estimates and testsfor multimodalityrdquo Journal of the American Statistical Associa-tion vol 86 no 415 pp 738ndash746 1991

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Decision SciencesAdvances in


AnalysisInternational Journal of


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Submit your manuscripts atwwwhindawicom

2 Complexity

(a) (b)

(c) (d)

Figure 1 Typical fission-fusion behaviors of flocking system in nature (a) European starlings Sturnus vulgaris over Rome [3] (b) shoal ofsardines in Kwazulu-natal South Africa (c) Merinos drsquoArles sheep Ovis aries [6] (d) segregation in mixed cocultures of primary goldfishkeratocytes (PFK red) and EPC fish keratocytes (EPC green) [18]

particles considering the alignment rule only was introducedin [11] In addition a zonal model was proposed in [12]which is able to produce some typical collective behaviors(such as swarm torus dynamic parallel and highly dynamicparallel group) by adjusting certain parameters In generalthese worksmainly deploy the egalitarian interaction methodand focus on the group fusion aspect little attention has beenpaid to the fission behaviorHow a randomly distributed flockforms a coherent group and splits into multiple subgroups ina unified manner remains largely unknown

Recently owing to the advancements in the high-resolution spatiotemporal flocking data acquisition and anal-ysis techniques more and more lines of evidence havedemonstrated that the selective interaction rather than theegalitarian interaction is more effective in the collectiveresponse of flocking system [13 14] A selective interactionbased hierarchical structure was found in pigeon flockrsquosdirectional choice which was proved to bemore flexible thanthe egalitarian strategy [15] In [16] a hybrid control architec-ture combining both compromise (egalitarian interaction)and leadership (selective interaction) was proposed to drivethe grouprsquos movement decisions A route-dependent switchmechanism between hierarchical and egalitarian strategies inpigeon flocks was found in [17] which revealed that pigeonstend to follow the average of neighbors while moving along asmooth trajectory and select a certain leader to follow whensudden turns or zigzags occurs Selective interaction whichimplements the information transfer by following a specificleader in an implicit manner shows significant potential inthe fission behavior of flocking system

Inspired by the above empirical evidences an intermit-tent selective interaction based control algorithm is proposedfor the self-organized fission-fusion behavior of flockingsystem The main contributions of this paper are as follows

(i) A specifically designed intermittent selective inter-action behavior is integrated into the conventionalaverage velocity consensus scheme and endows theflock the capability of spontaneous splitting in thepresence of external stimulus information

(ii) Aweight adjustment strategy is designed to automati-cally balance the egalitarian interaction and intermit-tent selective interaction in different environmentswhich guarantees the stability and environmentaladaptability of flocking system

(iii) A dynamic threshold value for the fission behavior isdesigned based on the order parameter of individualswhich make the flock more flexible in the groupfission-fusion behavior

The rest of this paper is organized as follows In Section 2the coordinated control problem for the self-organizedfission-fusion behavior of flocking system is proposed and thepitfall of the conventional velocity consensus based egalitar-ian interaction in flocking control is analyzed In Section 3a unified fission-fusion control framework is established byintroducing the intermittent selective interaction into theegalitarian interaction scheme and a dynamic weight balancestrategy is designed for the group fusion and fission behaviorNumerical simulations are provided in Section 4 to verify

Complexity 3




119909119894 = V119894

V119894 = 119906119894 (1)



119873119894 (119905) = 119895 119889119894119895 le 119877 119895 = 1 2 sdot sdot sdot 119873 119895 = 119894 (2)



119906119894 = 119906in119894 ( sum119895isin119873119894(119905)

119906 (119909119894 119909119895 V119894 V119895)) + 119892119894119906out119894 (3)




119906in119894 = 119906pos119894 + 119906vel119894 (4)


119906pos119894 = sum119895isin119873119894(119905)





(V119894 minus V119895) (6)







4 Complexity



+ 119892119894119906out119894 (7)





(V119894 minus V119895)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119906ega119894


(8)












119862119894119895 = 120577119894119895 110038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198601

∙ 120581119894119895 (V119894 ∙ V119895)1003817100381710038171003817V1198941003817100381710038171003817 10038171003817100381710038171003817V11989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198602

(10)



119862lowast = 119890minus120573120593119894 (11)


10038171003817100381710038171003817100381710038171003817100381710038171003817119873119894sum119895=1

V11989510038171003817100381710038171003817V1198951003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (12)



Complexity 5






120595 = 11198731003817100381710038171003817100381710038171003817100381710038171003817119873sum119894=1

V1198941003817100381710038171003817V11989410038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (13)




120582 = S2 + 1K

(14)

where












119878sum119895=1





6 Complexity



Value 20 3 05 2 25 05 123 15m 3m 55

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(a)

minus10

0

10

20

30

40

50

60

70

y (m

)0 10 20 30 40 50 60 70minus10

x (m)

(b)

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(c)

0 10 20 30 40 50 60 70minus10x (m)

minus10

0

10

20

30

40

50

60

70

y (m

)

(d)






Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y


20 6 84 10t (s)

0

05

1





minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)






8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3




119909119894 = V119894

V119894 = 119906119894 (1)



119873119894 (119905) = 119895 119889119894119895 le 119877 119895 = 1 2 sdot sdot sdot 119873 119895 = 119894 (2)



119906119894 = 119906in119894 ( sum119895isin119873119894(119905)

119906 (119909119894 119909119895 V119894 V119895)) + 119892119894119906out119894 (3)




119906in119894 = 119906pos119894 + 119906vel119894 (4)


119906pos119894 = sum119895isin119873119894(119905)





(V119894 minus V119895) (6)







4 Complexity



+ 119892119894119906out119894 (7)





(V119894 minus V119895)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119906ega119894


(8)












119862119894119895 = 120577119894119895 110038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198601

∙ 120581119894119895 (V119894 ∙ V119895)1003817100381710038171003817V1198941003817100381710038171003817 10038171003817100381710038171003817V11989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198602

(10)



119862lowast = 119890minus120573120593119894 (11)


10038171003817100381710038171003817100381710038171003817100381710038171003817119873119894sum119895=1

V11989510038171003817100381710038171003817V1198951003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (12)



Complexity 5






120595 = 11198731003817100381710038171003817100381710038171003817100381710038171003817119873sum119894=1

V1198941003817100381710038171003817V11989410038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (13)




120582 = S2 + 1K

(14)

where












119878sum119895=1





6 Complexity



Value 20 3 05 2 25 05 123 15m 3m 55

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(a)

minus10

0

10

20

30

40

50

60

70

y (m

)0 10 20 30 40 50 60 70minus10

x (m)

(b)

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(c)

0 10 20 30 40 50 60 70minus10x (m)

minus10

0

10

20

30

40

50

60

70

y (m

)

(d)






Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y


20 6 84 10t (s)

0

05

1





minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)






8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity



+ 119892119894119906out119894 (7)





(V119894 minus V119895)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119906ega119894


(8)












119862119894119895 = 120577119894119895 110038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198601

∙ 120581119894119895 (V119894 ∙ V119895)1003817100381710038171003817V1198941003817100381710038171003817 10038171003817100381710038171003817V11989510038171003817100381710038171003817⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198602

(10)



119862lowast = 119890minus120573120593119894 (11)


10038171003817100381710038171003817100381710038171003817100381710038171003817119873119894sum119895=1

V11989510038171003817100381710038171003817V1198951003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (12)



Complexity 5






120595 = 11198731003817100381710038171003817100381710038171003817100381710038171003817119873sum119894=1

V1198941003817100381710038171003817V11989410038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (13)




120582 = S2 + 1K

(14)

where












119878sum119895=1





6 Complexity



Value 20 3 05 2 25 05 123 15m 3m 55

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(a)

minus10

0

10

20

30

40

50

60

70

y (m

)0 10 20 30 40 50 60 70minus10

x (m)

(b)

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(c)

0 10 20 30 40 50 60 70minus10x (m)

minus10

0

10

20

30

40

50

60

70

y (m

)

(d)






Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y


20 6 84 10t (s)

0

05

1





minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)






8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5






120595 = 11198731003817100381710038171003817100381710038171003817100381710038171003817119873sum119894=1

V1198941003817100381710038171003817V11989410038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 (13)




120582 = S2 + 1K

(14)

where












119878sum119895=1





6 Complexity



Value 20 3 05 2 25 05 123 15m 3m 55

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(a)

minus10

0

10

20

30

40

50

60

70

y (m

)0 10 20 30 40 50 60 70minus10

x (m)

(b)

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(c)

0 10 20 30 40 50 60 70minus10x (m)

minus10

0

10

20

30

40

50

60

70

y (m

)

(d)






Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y


20 6 84 10t (s)

0

05

1





minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)






8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity



Value 20 3 05 2 25 05 123 15m 3m 55

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(a)

minus10

0

10

20

30

40

50

60

70

y (m

)0 10 20 30 40 50 60 70minus10

x (m)

(b)

minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)

(c)

0 10 20 30 40 50 60 70minus10x (m)

minus10

0

10

20

30

40

50

60

70

y (m

)

(d)






Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y


20 6 84 10t (s)

0

05

1





minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)






8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v x

0 1 2 3 4 5 6 7 8minus5

0

5

10

15

t (s)

v y


20 6 84 10t (s)

0

05

1





minus10

0

10

20

30

40

50

60

70

y (m

)

0 10 20 30 40 50 60 70minus10x (m)






8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

minus10

0

10

20

v x

minus10

0

10

20

v y

1 2 3 4 5 6 7 80t (s)

1 2 3 4 5 6 7 80t (s)


0

05

1

2 4 6 8 100t (s)







+ 119906nav119894 + 119892119894119906out119894 (19)


0

10

20

30

40

50

60

70

80

90

y (m

)

10 20 30 40 50 60 70 80 900x (m)






Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9

=1=07=05

=03=01=0

0

01

02

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)





=005=015=035

=05=075=1

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity

=1=2=4

=6=8=10

03

04

05

06

07

08

09

1

1 2 3 4 5 6 7 8 9 100t (s)






=01=05=1

=2=5=10

1 2 3 4 5 6 7 8 9 100t (s)

0

01

02

03

04

05

06

07

08

09

1


1 2 3 4 5 6 7 8 9 100t (s)

04

05

06

07

08

09

1

=01=05=1

=2=5=10




Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11







Data Availability




Acknowledgments


References

















12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Self-Organized Fission-Fusion Control Algorithm for Flocking …Complexity T ˘ˇ : Parametersinthesimulation. Parameters ˆ˝ $ tar& - C " esc tar F esc Value . . . . . m m . −10